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Find the number of elements in the cyclic subgroup of  $\mathbb{Z_{30}}$ generated by $25$.

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Cyclic Subgroups :

If we pick some element a from a group G then we can consider the subset of all elements of G that are powers of a. This subset forms a subgroup of G and is called the cyclic subgroup generated by a. If forms a subgroup since it is

• Closed. If you multiply powers of a you end up with powers of a
• Has the identity. $a \ast a^{-1} = a^{0} = e$

For $\mathbb{Z_{30}}$( for additive operation),

$25+ 25= 50= 20 (\mod 30).$

$25+ 25+ 25= 75= 15 (\mod 30)$.

$25+ 25+ 25+ 25= 100= 10 (\mod 30)$.

$25+ 25+ 25+ 25+ 25= 125= 5 (\mod 30).$

$25+25+ 25+ 25+ 25+ 25= 150= 0 (\mod 50).$

$25+25+ 25+ 25+ 25+ 25+ 25= 175= 25 (\mod 30).$

There are $6$ such groups.

Is there any better approach ?